Standard part function

In nonstandard analysis, the standard part function is a function from the limited (finite) hyperreal numbers to the real numbers.

Briefly, the standard part function "rounds off" a finite hyperreal to the nearest real.

As such, it is a mathematical implementation of the historical concept of adequality introduced by Pierre de Fermat,[1] as well as Leibniz's Transcendental law of homogeneity.

The standard part function was first defined by Abraham Robinson who used the notation

for the standard part of a hyperreal

This concept plays a key role in defining the concepts of the calculus, such as continuity, the derivative, and the integral, in nonstandard analysis.

The latter theory is a rigorous formalization of calculations with infinitesimals.

The standard part of x is sometimes referred to as its shadow.

[2] Nonstandard analysis deals primarily with the pair

are an ordered field extension of the reals

, and contain infinitesimals, in addition to the reals.

In the hyperreal line every real number has a collection of numbers (called a monad, or halo) of hyperreals infinitely close to it.

The standard part function associates to a finite hyperreal x, the unique standard real number x0 that is infinitely close to it.

The relationship is expressed symbolically by writing The standard part of any infinitesimal is 0.

Thus if N is an infinite hypernatural, then 1/N is infinitesimal, and st(1/N) = 0.

is represented by a Cauchy sequence

in the ultrapower construction, then More generally, each finite

defines a Dedekind cut on the subset

) and the corresponding real number is the standard part of u.

The standard part function "st" is not defined by an internal set.

Perhaps the simplest is that its domain L, which is the collection of limited (i.e. finite) hyperreals, is not an internal set.

Alternatively, the range of "st" is

[3] All the traditional notions of calculus can be expressed in terms of the standard part function, as follows.

The standard part function is used to define the derivative of a function f. If f is a real function, and h is infinitesimal, and if f′(x) exists, then Alternatively, if

, one takes an infinitesimal increment

The derivative is then defined as the standard part of the ratio: Given a function

as the standard part of an infinite Riemann sum

is taken to be infinitesimal, exploiting a hyperfinite partition of the interval [a,b].

Here the limit is said to exist if the standard part is the same regardless of the infinite index chosen.

is continuous at a real point